Transcript P- and S-wave velocities, and sensor location depths

FUNDAMENTALS of
ENGINEERING SEISMOLOGY
GROUND MOTIONS
FROM SIMULATIONS
1
Observed data adequate for
regression except
close to large ‘quakes
Observed data not adequate for
regression, use simulated data
2
Ground-Motions for Regions Lacking Data
from Earthquakes in Magnitude-Distance
Region of Engineering Interest
• Most predictions based on the stochastic method,
using data from smaller earthquakes to constrain
such things as path and site effects
Simulation of ground motions
• representation theorem
• source
– dynamic
– kinematic
• path & site
– wave propagation
– represent with simple functions
Representation Theorem

u n x , t  
 d  u  ,  c


i

j
ijpq

Gnp



x , t   ;  ,0
 q
Computed ground displacement
5
 d
Representation Theorem

u n x , t  
 d  u  ,  c


i


j
ijpq
Gnp



x , t   ;  ,0
 q
Model for the slip on the fault
6
 d
Representation Theorem

u n x , t  
 d  u  ,  c


i

j
Gnp

ijpq



x , t   ;  ,0
 q
Green’s function
7
 d
Types of simulations
• Deterministic
–
–
–
–
Purely theoretical
Deterministic description of source
Wave propagation in layered media
Used for lower frequency motions
• Stochastic
–
–
–
–
–
Not purely theoretical
Random source properties
Capture wave propagation by simple functional forms
Can use deterministic calculations for some parts
Primarily for higher frequencies (of most engineering
concern)
Types of simulations
• Hybrid
– Meaning 1: Deterministic at low frequencies,
stochastic at high frequencies
– Meaning 2: Combine empirical ground-motion
prediction equations with stochastic simulations to
account for differences in source and path
properties (Campbell, ENA).
– Meaning 3: Stochastic source, empirical Green’s
function for path and site
Stochastic simulations
• Point source
– With appropriate choice of source scaling, duration,
geometrical spreading, and distance can capture some
effects of finite source
• Finite source
– Many models, no consensus on the best (blind prediction
experiments show large variability)
– Usually use point-source stochastic model
– Can be theoretical (many types: deterministic and/or
stochastic, and can also use empirical Green’s functions)
The stochastic method
• Overview of the stochastic method
– Time-series simulations
– Random-vibration simulations
• Target amplitude spectrum
– Source: M0, f0, Ds, source duration
– Path: Q(f), G(R), path duration
– Site: k, generic amplification
• Some practical points
Stochastic modelling of ground-motion:
Point Source
• Deterministic modelling of high-frequency waves not
possible (lack of Earth detail and computational
limitations)
• Treat high-frequency motions as filtered white noise
(Hanks & McGuire , 1981).
• combine deterministic target amplitude obtained from
simple seismological model and quasi-random phase
to obtain high-frequency motion. Try to capture the
essence of the physics using simple functional forms
for the seismological model. Use empirical data
when possible to determine the parameters.
12
Basis of stochastic method
Radiated energy described by
the spectra in the top graph is
assumed to be distributed
randomly over a duration given
by the addition of the source
duration and a distantdependent duration that
captures the effect of wave
propagation and scattering of
energy
These are the results of actual
simulations; the only thing that
changed in the input to the
computer program was the
moment magnitude (4, 6, and 8)
13
Acceleration (cm/s 2)
200
-200
Tim e: 22:16:44
Velocity (cm/sec)
10
Response
1000
of Wood-Anderson
seismometer
Date: 2003-09-18;
-10
(cm)
-1000
20
0
-20
Acceleration (cm/s 2)
0.5
Velocity (cm/sec)
File: C:\m etu_03\sim ulation\AS47WA_4ppt.draw;
Acceleration,
velocity,
oscillator
response for
two very
different
magnitudes,
changing
only the
magnitude in
the input file
M=7,R=10
M=4,R=10
0
-0.5
5
0
-5
Response
0
10
of Wood-Anderson
20
Time (sec)
seismometer
30
14
(cm)
40
• Ground motion and response parameters can be
obtained via two separate approaches:
– Time-series simulation:
• Superimpose a quasi-random phase spectrum on a
deterministic amplitude spectrum and compute synthetic
record
• All measures of ground motion can be obtained
– Random vibration simulation:
• Probability distribution of peaks is used to obtain peak
parameters directly from the target spectrum
• Very fast
• Can be used in cases when very long time series, requiring
very large Fourier transforms, are expected (large
distances, large magnitudes)
• Elastic response spectra, PGA, PGV, PGD, equivalent
linear (SHAKE-like) soil response can be obtained
15
“The Stochastic method has a long history of performing better than
it should in terms of matching observed ground-motion
characteristics. It is a simple tool that combines a good deal of
empiricism with a little seismology and yet has been as successful
as more sophisticated methods in predicting ground-motion
amplitudes over a broad range of magnitudes, distances,
frequencies, and tectonic environments. It has the considerable
advantage of being simple and versatile and requiring little advance
information on the slip distribution or details of the Earth structure.
For this reason, it is not only a good modeling tool for past
earthquakes, but a valuable tool for predicting ground motion for
future events with unknown slip distributions.”
--Motazedian and Atkinson (2005)
16
Time-domain simulation
17
Step 1: Generation of random white
noise
• Aim: Signal with random
phase characteristics
• Probability distribution for
amplitude
– Gaussian (usual choice)
– Uniform
• Array size from
– Target duration
Tgm  Tsource  Tpath
– Time step (explicit input
parameter)
18
Step 2: Windowing the noise
• Aim: produce time-series
that look realistic
• Windowing function
– Boxcar
– Cosine-tapered
boxcar
– Saragoni & Hart
(exponential)
19
Step 3: Transformation to frequency-domain
• FFT algorithm
20
Step 4: Normalisation of noise
spectrum
• Divide by rms integral
• Aim of random noise
generation = simulate
random PHASE only
Normalisation required to keep energy content
dictated by deterministic amplitude spectrum
21
Step 5: Multiply random noise spectrum by
deterministic target amplitude spectrum
Normalised
amplitude spectrum
of noise with
random phase
characteristics
Y(M0,R,f) =
TARGET
FOURIER
AMPLITUDE
SPECTRUM
E(M0,f)
Earthquake
source
X
P(R,f)
Propagation
path
X
G(f)
Site
response
X
I(f)
Instrument or
ground
motion
Step 6: Transformation back to timedomain
• Numerical IFFT yields
acceleration time series
• Manipulation as with
empirical record
• 1 run = 1 realisation
of random process
– Single time-history not
necessarily realistic
– Values calculated =
average over N simulations
(N large enough to yield an
accurate value of groundmotion intensity measure)
23
Steps in simulating time series
• Generate Gaussian or uniformly
distributed random white noise
• Apply a shaping window in the
time domain
• Compute Fourier transform of the
windowed time series
• Normalize so that the average
squared amplitude is unity
• Multiply by the spectral amplitude
and shape of the ground motion
• Transform back to the time
domain
24
100
box window
exponential
window
Effect of shaping
window on
response spectra
M = 7, R = 10 km
Tim e: 21:47:27
Date: 2003-09-18;
200
Using box window
0
2
-200
200
Using exponential
window
0
-200
20
30
Time (sec)
40
50
1
0.1
1
Period (sec) 25
10
File: C:\m etu_03\sim ulation\box_exp_4ppt.draw;
10
Acceleration (cm/s 2)
psv (cm/s)
20
Warning: the
spectrum of any
one simulation may
not closely match
the specified
spectrum. Only the
average of many
simulations is
guaranteed to
match the specified
spectrum
26
Random Vibration
Simulation
27
Random Vibration Simulations General
• Aim:
Improve efficiency by using Random
Vibration Theory to model random phase
• Principle:
– no time-series generation
– peak measure of motion obtained directly
from deterministic Fourier amplitude
spectrum through rms estimate
• yrms is easy to obtain from amplitude spectrum:
 yrms 
2
1

Drms
Td

2
2
0 u(t ) dt  Drms 0 U ( f ) df
2
yrms
is root-mean-square motion
u(t )
Is ground-motion time series (e.g.,
accel. or osc. response)
Parseval's
theorem
Drms is a duration measure
2
U ( f ) is Fourier amplitude spectrum of
ground motion
• But need extreme value statistics to relate rms acceleration to peak
time-domain ground-motion intensity measure (ymax)
29
Peak parameters from random vibration theory:
For long duration (D) this equation gives the peak motion given the
rms motion:
ymax
12
  2 ln N Z 
yrms
where
NZ  2 f Z D
1
12
fZ 
 m2 m0 
2
m0 and m2 are spectral moments, given by integrals over the
Fourier spectra of the ground motion
30
Random Vibration Simulation – Possible
Limitations
• Neither stationarity nor uncorrelated peaks assumption true
for real earthquake signal
• Nevertheless, RV yields good results at greatly reduced
computer time
• Problems essentially with
– Long-period response
– Lightly damped oscillators
– Corrections developed
31
Special consideration
needs to be given to
choosing the proper
duration T to be used in
random vibration theory
for computing the
response spectra for
small magnitudes and
long oscillator periods.
In this case the oscillator
response is short
duration, with little
ringing as in the
response for a larger
earthquake. Several
modifications to rvt have
been published to deal
with this.
27
acceleration(low -cutfiltered, fc=0.1 Hz)
M= 4
-21
0.01
T=10 sec, 5% damped oscillatorr esponse
-0.04
420
acceleration(low -cutfiltered, fc=0.1 Hz)
-510
13
M= 7
T=10 sec,5% dampedo scillatorr esponse
-13
02
04
06
08
time
Dec1 6, 2000 9:08:37a m
C:\AKI_SYMP\RDT SM4M7.GRA
C:\AKI_SYMP\RDT SM4M7.DT
32
0
100
2
Tim e: 10:46:42
1
Date: 2003-09-19;
0.2
0.1
For M = 4, R = 10
km
0.02
File: C:\m etu_03\sim ulation\nsim s_m 4_4ppt.draw;
psv (cm/s)
Comparison of
time domain and
random vibration
calculations,
using two
methods for
dealing with
nonstationary
oscillator
response.
M = 4.0, R = 10 km
Random Vibration, Boore & Joyner
Random Vibration, Liu & Pezeshk
Time Domain: 10 runs
Time Domain: 40 runs
Time Domain: 160 runs
Time Domain: 640 runs
0.01
0.1
1
Period (sec)
10
33
100
Date: 2003-09-19;
20
10
File: C:\m etu_03\sim ulation\nsim s_m 7_4ppt.draw;
psv (cm/s)
Comparison of
time domain and
random vibration
calculations,
using two
methods for
dealing with
nonstationary
oscillator
response.
Tim e: 10:57:27
M = 7.0, R = 10 km
Random Vibration, Boore & Joyner
Random Vibration, Liu & Pezeshk
Time Domain: 10 runs
Time Domain: 40 runs
Time Domain: 160 runs
Time Domain: 640 runs
For M = 7, R = 10
km
2
1
0.1
1
Period (sec)
10
34
Recent improvements
on determining Drms
(Boore and
Thompson, 2012):
Contour plots of
TD/RV ratios for an
ENA SCF 250 bar
model for 4 ways of
determining Drms:
1.
2.
3.
4.
Drms = Dex
BJ84
LP99
BT12
35
Target amplitude spectrum
Deterministic function of source, path and site
characteristics represented by separate multiplicative filters
Y (M0 , R, f )  E(M 0 , f )  P(R, f )  G( f )  I ( f )
Earthquake
source
Instrument
Propagation
Site
path
response or ground
motion
THE KEY TO THE SUCCESS OF THE MODEL LIES IN BEING
ABLE TO DEFINE FOURIER ACCELERATION SPECTRUM
AS F(M, DIST)
Parameters required to specify Fourier accn as
f(M,dist)
•
•
•
•
•
Model of earthquake source spectrum
Attenuation of spectrum with distance
Duration of motion [=f(M, d)]
Crustal constants (density, velocity)
Near-surface attenuation (fmax or kappa)
Stochastic method
• To the extent possible the spectrum is
given by seismological models
• Complex physics is encapsulated into
simple functional forms
• Empirical findings can be easily
incorporated
Source Function
39
Source function E(M0, f)
E(M 0 , f )  C  M 0  S (M 0 , f )
Scaling constant
Seismic moment
• near-source crustal properties
• assumptions about wave-type
considered (e.g. SH)
Measure of
earthquake size
Source
DISPLACEMENT
Spectrum
Scaling of amplitude
spectrum with
earthquake size
Scaling constant C (frequency
independent)
•βs = near-source shear-wave velocity
C
R VF
4 s  R0
3
s
•ρs = near-source crustal density
•V = partition factor
•(Rθφ) = average radiation pattern
•F = free surface factor
•R0 = reference distance (1 km).
Brune source model
• Brune’s point-source model
– Good description of small, simple ruptures
– "surprisingly good approximation for many large events". (Atkinson &
Beresnev 1997)
• Single-corner frequency model
S( f ) 
1
1

f 2
2
1 2
1 2
f0
0
• High-frequency amplitude of acceleration scales as:
ahf  M 01 3 2 3
Semi-empirical two-corner-frequency models
• Aim: incorporate finite-source effects by refining the
source scaling
• Example: AB95 & AS00 models
1 

S( f ) 

2
f
f2
1 2
1 2
fa
fb
• Keep Brune's HF amplitude scaling
ahf  M
13
0

23
fa, fb and e determined
empirically (visual
inspection & best-fit)
Stress parameter: definitions
• "Stress drop" should be reserved for static measure of
slip relative to fault dimensions
u
 
r
u = average slip
r = characterisic fault dimension
• "Brune stress drop" = change in tectonic (static) stress
due to the event
• "SMSIM stress parameter" = “parameter controlling
strength of high-frequency radiation” (Boore 1983)
ahf  M
13
0

23
Stress parameter - Values
• SMSIM stress parameter
– California: 50 - 200 bar
– ENA: 150 – 1000 bar
(greater uncertainty)
45
Tim e: 14:49:29
10000
Date: 2003-09-15;
M = 7.5
1000
100
File: C:\m etu_03\rec_proc_strong_m otion\FAS_XCA.draw;
Fourier Acceleration Spectrum (cm/s)
The spectra can be more
complex in shape and
dependence on source
size. These are some of
the spectra proposed
and used for simulating
ground motions in
eastern North America.
The stochastic method
does not care which
spectral model is used.
Providing the best model
parameters is essential
for reliable simulation
results (garbage in,
garbage out).
M = 4.5
10
AB95
H96
Fea96 (no site amp)
BC92
J97
1
0.1
0.01
0.1
1
Frequency (Hz)
46
10
100
•High frequency
•A ≈ M0(1/3), but log M0 ≈
1.5M, so A ≈ 100.5M. This
is a factor of 3 for a unit
increase in M
•Ground motion at
frequencies of engineering
interest does not increase by
10x for each unit increase in
M
M0 f03 = constant (similarity: Aki, 1967)
)(2 / 3)
10 26
M0
10 25
-square spectra: Aki (1967)
M 7.5
= 100 bars
= 200 bars
• The key is to describe how
the corner frequencies vary
with M. Even for more
complex sources, often try to
relate the high-frequency
spectral level to a single
stress parameter
(M 0)(1 / 3) (
Date: 2003-09-15;
Acceleration Source Spectrum
•A≈ M0, but log M0 ≈ 1.5M,
so A ≈ 101.5M. This is a
factor of 32 for a unit
increase in M
10 27
File: C:\metu_03\rec_proc_strong_motion\SPCTSCL_aki_fig2_4ppt.draw;
•Low frequency
Time: 14:58:22
Source Scaling
M 6.5
10 24
0.01
0.10
1.0
Frequency (Hz)
10
47
Source duration
• Determined from source scaling model via:
Tsource 
wfa
fa

w fb
fb
– For single-corner model, fa= fb = f0
Path Function
49
Path function P(R, f)
P(R,f)
=
Geometrical
Spreading (R)
Point-source => spherical wave
Loss of energy
through spreading
of the wavefront
Anelastic
Attenuation (R,f)
Propagation medium is
neither perfectly elastic nor
perfectly homogeneous
Loss of energy through
material damping &
wave scattering by
heterogeneities
Wave propagation
produces changes
of amplitude (not
shown here) and
increased duration
with distance. Not
included in these
simulations is the
effect of
scattering, which
adds more
increases in
duration and
smooths over
some of the
amplitude
changes (as does
combining
propagation over
various profiles
with laterally
changing velocity)
51
The overall behavior of
complex path-related effects
can be captured by simple
functions, leaving aleatory
scatter. In this case the
observations can be fit with a
simple geometrical spreading
and a frequency-dependent Q
operator
52
1
0
0
-1
-1
-2
-2
-3
-3
3.0 <= mbLg <= 3.4
vertical component
Tim e: 10:22:23
f = 16 Hz
Date: 2003-09-18;
f = 1.0 Hz
File: C:\m etu_03\sim ulation\Fas01_16_4ppt.draw;
log Fourier amplitude (cm/sec)
1
3.0 <= mbLg <= 3.4
vertical component
-4
-4
1
1.5
2
2.5
log distance (km)
3
1
1.5
2
2.5
log distance (km)
In eastern North America a more complicated geometrical
spreading factor is needed (here combined with the Q
operator)
53
3
Geometrical Spreading Function
• Often 1/R decay (spherical wave), at
least within a few tens of km
• SMSIM allows n segment piecewise
linear function in log (amplitude) –
log (R) space
• Magnitude-dependent slopes possible
: allows to capture finite-source effect
(Silva 2002)
Tim e: 10:24:17
Date: 2003-09-18;
File: C:\m etu_03\sim ulation\GSPRDFIG.draw;
1/R
Geometrical Spreading
• At greater distances, the decay is
better characterised by 1/Ra with a
<1
0.1
0.03
0.02
1/70
0.01
1/70 (130/R)
10
20
30
100
0.5
200
300
Distance (km)
Example: Boore & Atkinson 1995 Eastern
North America model
54
Path-dependent anelastic attenuation
Form of filter:
exp(
π f R
Q ( f )cq
)
– Q(f) = « quality factor » of propagation medium in terms
of (shear) wave transmission
– cq= velocity used to derive Q; often taken equal to s
(not strictly true – depends on source depth)
– N.B. Distance-independent term (kappa) removed since
accounted for elsewhere
55
0.1
n
Hi
du
Ku
sh
Tim e: 10:21:39
pe
lifo
l
ria
rn
u
Fa
ia
lt,
Date: 2003-09-18;
0.01
Im
Ca
ifo
al
C
File: C:\m etu_03\sim ulation\aki_q_4ppt.draw;
Q-1
ia
rn
ska
Ala
Ga
0.001
rm
,C
ent
ral
As
ia
Surface
Waves
So
uth
Ku
rile
Cen
tral
U.S.
10 -4
0.01
0.1
1
Frequency (Hz)
10
100
Observed Q from around the world, indicating general
dependence on frequency
56
Wave-transmission quality factor Q(f)
• Form usually assumed:
n
pe
=
s2
pe
=s
1
0.001
ft2
File: C:\metu_03\sim ulation\IQ_FIG_4ppt.draw;
ft1
Date: 2003-09-18;
slo
(fr1, Qr1)
Time: 10:23:12
(fr2, Qr2)
0.01
slo
• Study of published relations
led Boore to assign 3segment piecewise linear
form (in log-log space)
Q-1
Q ( f )  Q0  f f 0 
0.1
10 -4
0.01
0.1
1
Frequency (Hz)
57
10
100
Empirical determination of Q(f)
• Assuming simple functional forms for source
function and geometrical spreading function (e.g.
Brune model & 1/R decay)
– Source-cancelling (for constant Q)
– Best fit for assumed functional form (Q=Q0fn)
• Simultaneous inversion of source, path and site
effects
– One unconstrained dimension => additional assumption
required (e.g. site amplification)
• Trade-off problems
Path duration
• Required for array size
observed
30
• Usually assumed linear
with distance
25
Tim e: 10:35:40
Date: 2003-09-18;
20
15
File: C:\m etu_03\sim ulation\DURN2_4ppt.draw;
• Regional characteristic,
should be determined from
empirical data
duration (sec)
• SMSIM allows
representation by a
piecewise linear function
- source
mean in 15-km wide bins
used by Atkinson & Boore (1995)
10
5
• Example: AB95 for ENA
0
0
100
200
300
distance (km)
59
400
500
Site Response Function
60
Site response
• Form of filter:
G( f )  A( f )  D( f )
Linear
amplification for
GENERIC site
Regional distance-independent
attenuation (high frequency)
• Near-surface anelastic attenuation?
• Source effect?
• Combination?
Site amplification
• Attenuation function for
GENERIC site
• Modelled as a piecewise linear
function in log-log space
• Soil non-linearity effects not
included
• Determined from crustal
velocity & density profile via
SITE_AMP
– Square-root of impedance
approximation
– Quarter-wave-length approximation
(f – dependent)
62
Site attenuation
• Form of filter:
D( f ) 
1
 f
1  
 f max



8
exp(k 0 f )
• Reflects lack of consensus about representation
– fmax (Hanks, 1982) : high-frequency cut-off
– k0 (Anderson & Hough, 1984) : high-frequency decay
– Both factors seldom used together
Cut-off frequency fmax
• Hanks (1982)
– Observed empirical spectra exhibit cut-off in log-log space
– Value of cut-off in narrow range of frequency
– Attributed to site effect
• Other authors (e.g. Papageorgiou & Aki 1983) consider
fmax to be a source effect
• Boore’s position:
– Multiplicative nature of filter allows for both approaches
– Classification as site effect = « book-keeping » matter
– Often set to a high value (50 to 100 Hz) when preference is given
to the kappa filter
Kappa factor k0
• Anderson & Hough (1984)
– empirical spectra plotted in semi-log axes exhibit exponential HF
decay
– rate of this decay = k varies with distance)
• Treatment in SMSIM
– similar determination, but with records corrected for path effects
and site amplification
– parameter used = k0 = zero-distance intercept
– allowed to vary with magnitude
• source effect (at least partly)
• trade-off with source strength (characteristic of regional surface
geology)
Combined effect of generic rock amplification and diminution
(diminution = e(- 0 f))
4
0
2
0
0
0
Tim e: 10:45:55
= 0.0 s
Date: 2003-09-18;
0
3
= 0.01
= 0.02
File: C:\m etu_03\sim ulation\A_K4PAPR_4ppt.draw;
Amplification, relative to source
Site response is
represented by a
table of frequencies
and amplifications.
Shown here is the
generic rock
amplification for
coastal California (k
= 0.04), combined
with the k
diminution factor for
various values of k
= 0.04
= 0.08
1
0.1
1
10
Frequency (Hz)
66
100
4
0
= 0.035 s, except = 0.003 s for very hard rock
3
1
V30
V30
V30
V30
V30
0.4
0.01
0.1
= 255 m/s (NEHRP class D)
= 310 m/s (generic soil)
= 520 m/s (NEHRP class C)
= 620 m/s (generic rock)
= 2900 m/s (generic very hard rock)
1
Frequency (Hz)
10
67
File: C:\metu_03\sim ulation\AMPK_FAS_4ppt.draw;
Date: 2003-09-18;
0f)
Time: 11:31:37
2
Amps*e(-
Combined
amplification
and diminution
filter for various
average site
classes
Instrument/Ground motion filter
• For ground-motion simulations:
I ( f )  (2 f i) n
i² = 1
n=0 displacement
n=1 velocity
n=1 acceleration
• For oscillator response:
 Vf 2
I( f ) 
2
2
( f  f r )  2iff r 
fr = undamped natural frequency
ξ = damping
V = gain
(for response spectra, V=1).
Summary of Parameters Needed
for Stochastic Simulations
69
Parameters needed for Stochastic
Simulations
• Frequency-independent parameters
– Density near the source
– Shear-wave velocity near the source
– Average radiation pattern
– Partition factor of motion into two components
(usually 1 2 )
– Free surface factor (usually 2)
70
Parameters needed for Stochastic
Simulations
• Frequency-dependent parameters
– Source:
• Spectral shape (e.g., single corner frequency; two
corner frequency)
• Scaling of shape with magnitude (controlled by the
stress parameter Δσ for single-corner-frequency
models)
71
Parameters needed for Stochastic
Simulations
Frequency-dependent parameters
– Path (and site):
• Geometrical spreading (multi-segments?)
• Q (frequency-dependent? What shear-wave and
geometrical spreading model used in Q determination?)
• Duration
• Crustal amplification (can include local site
amplification)
• Site diminution (fmax? κ0?)
correlated
72
Parameters needed for Stochastic
Simulations
• RV or TD parameters
– Low-cut filter
– RV
• Integration parameters
• Method for computing Drms
• Equation for ymax/yrms
– TD
• Type of window (e.g., box, shaped?)
• dt, npts, nsims, etc.
73
Parameters that might be obtained from
empirical analysis of small earthquake data
• Focal depth distribution
• Crustal structure
– S-wave velocity profile
– Density profile
• Path Effects
–
–
–
–
–
Geometrical spreading
Q(f)
Duration
κ0
Site characteristics
74
Parameters difficult to obtain from small
earthquake data
• Source Spectral Shape
• Scaling of Source Spectra
(including determination of
Δσ)
75
Some points to consider
• Uncertainty
Specify parametric uncertainty when giving an
estimate from empirical data
• Trade-offs between parameters
–  and k0 (or similar)
– Geometrical spreading & anelastic attenuation
– Site amplification and k0
always state assumptions
• Consistency more important than
uniqueness
76
Applicability of Point Source Simulations
near Extended Ruptures
• Modify the value of Rrup used in point source,
to account for finite fault effects
– Use Reff (similar to Rrms) for a particular sourcestation geometry)
– Use a more generic modification, based on finitefault modeling (e.g., Atkinson and Silva, 2000;
Toro, 2002)
77
Use a
scenariospecific
modification
to Rrup
(note: No directivity---EXSIM results are an average of motions
from 100 random hypocenters)
Modified from Boore (2010)
78
Using generic modifications to
Rrup. For the situation in the
previous slide (M 7, Rrup =
2.5):
Reff = 10.3 km for AS00
Reff = 8.4 km for T02
Compared to Reff = 10.3 (off
tip) and 6.7 (normal) in the
previous slide
79
A few applications
• Scaling of ground motion with magnitude
• Simulating ground motions for a specific M, R
• Extrapolating observed ground motion to part
of M, R space with no observations
Other applications of the stochastic method
• Generate motions for many M, R; use
regression to fit ground-motion prediction
equations (used in U.S. National Seismic
Hazard Maps)
• Design-motion specification for critical
structures
• Derive parameters such as , k, from
strong-motion data
Other applications of the stochastic method
• Studies of sensitivity of ground motions to
model parameters
• Relate time-domain and frequency-domain
measures of ground motion
• Generate time series for use in nonlinear
analysis of structures and site response
• Use to compute motions from subfaults in
finite-fault simulations
Tim e: 12:19:58
T = 1.0 s
Date: 2003-09-19;
R = 20 km
100
File: C:\m etu_03\sim ulation\m ag_scaling_t0p1_0p3_1p0_2p0.draw;
Note that response
of long period
oscillators is more
sensitive to
magnitude than
short period
oscillators, as
expected from
previous
discussions. Also
note the nonlinear
magnitude
dependence for the
longer period
oscillators.
T = 2.0 s
1000
PSA(M)/(PSA(4.0)
Application: study
magnitude scaling
for a fixed distance
T = 0.3 s
10
T = 0.1 s
1
4
5
6
7
M
83
8
4
4
3
1
3
2
4
5
6
M
7
1
8
5
5
4
4
3
event_term
6
3
2
1
Date: 2005-05-24; Time: 10:49:37
5
4
5
6
M
7
8
4
5
6
M
7
8
2
1
4
5
6
M
7
8
0
T=0.1, 0.3, 1.0, 2.0 sec
rjb <
_ 80: buried events
rjb <
_ 80: events with surface slip
84
File: C:\peer_nga\teamx\stage1_event_terms_all_per_nobs_gt_1_rle_80_surfslip_smsim.draw;
5
event_term
6
2
event_term
Expected scaling
for simplest selfsimilar model.
Does this imply a
breakdown in self
similarity? The
simulations were
done for a fixed
close distance, and
more simulations
are needed at other
distances to be
sure that a Mdependent depth
term is not
contributing.
event_term
Magnitude scaling
6
Applications:
1000
Note comparison
with empirical
prediction equations
and importance of
basin waves (not in
simulations)
to 216 m/s
Date: 2003-09-19;
Time: 12:32:55
100
M 7.5
10
File: C:\metu_03\sim ulation\SEMS2AKI_4ppt.draw;
2) predict response
spectra for two
magnitudes without
use of the observed
recording
Observed (M 5.6)
Observed (M 5.6), scaled to M 7.5 using SMSIM
Observed, from S wave only (no basin waves)
Regression: Boore etal, 1997, V30 = 216 m/s
Regression: Abrahamson & Silva, 1997, corrected
Stochastic-method
simulation: AB98 model
PSV (cm/sec)
1) extrapolating
small earthquake
motion to large
earthquake motion
(makes use of path
and site effects in
the small ‘quake
recording)
M 5.6
1
0.1
0.1
0.2
1
2
Period (sec)
85
10
20
8
Western North America
Observed data adequate
for regression except
close to large ‘quakes
6
5
Used by BJF93 for pga
1
10
100
1000
8
Eastern North America
Moment Magnitude
7
6
5
Accelerographs
Seismographic Stations
1
10
100
Distance (km)
1000
File: C:\metu_03\regress\m_d_wna_ena_pga.draw; Date: 2003-09-05;Time: 14:40:01
Moment Magnitude
7
Observed data not
adequate for regression,
use simulated data
This is a major
application of
stochastic
simulations
86
Some observations about predicting ground motions
•
•
•
Empirical analysis:
– Adequate data in some places for M around 7
– Lacking data at close distances to large earthquakes
– Lacking data in most parts of the world
– May be possible to “export” relations from data-rich areas to
tectonically similar regions
– Need more data to resolve effects such as nonlinear soil response,
breakdown of similarity in scaling with magnitude, directivity, fault
normal/fault parallel motions, style of faulting, etc.
Stochastic method:
– Very useful for many applications
– Source specification for one region might be used for predictions in
another region
– Can use motions from more abundant smaller earthquakes to
define path, site effects
– Site effects, including nonlinear response, might be exportable from
data-rich regions
Hybrid method combining empirical and theoretical predictions can be
very useful
Some Limitations to Point-Source Model
(“A kindergarten model”—A. Frankel)
•
•
•
•
•
Faults are not points
No spatial coherence
No correlation between components
Inadequate period-to-period correlation
Etc.
Finite Faults
• Stochastic
–
–
–
–
–
–
–
–
Silva
Zeng et al.
Papageorgiou & Aki
Beresnev & Atkinson (FINSIM)
Motazedian & Atkinson (EXSIM)
Pacor, Cultrera, Mendez, & Cocco (DSM)
Irikura et al.
Etc.
• Deterministic + Stochastic
– Graves et al.
– Frankel
– Pacor et al. ? (simple Green’s function, does not include wave propagation
effects)
– Etc.
Extension of
stochastic model
to finite faults
(Silva; Beresnev
and Atkinson;
Motazedian and
Atkinson, Pacor et
al.)
90
Parameters needed to apply stochastic finitefault model
• All parameters needed for stochastic point source
model
• Geometry of source (can assume fault plane based
on empirical relations such as Wells and
Coppersmith on fault length and width vs. M)
• Direction of rupture propagation (can assume
random or bilateral)
• Slip distribution on fault (can assume random)
• Can also incorporate ‘self-healing’ slip behaviour
Summary
• Stochastic models are a useful tool for interpreting
ground motion records and developing ground
motion relations
• Model parameters need careful validation
• Point-source models appropriate for small
magnitudes, and work pretty well on average for
large magnitudes IF an adjustment is made to the
distance (and perhaps to the shape of the source
spectrum
• Finite-fault models preferable for large
earthquakes (M>6) because they can model more
realistic fault behaviour
Case Study
• M scaling-go over each input. Don’t forget
FFF, Rjb—Rrup calculation, discuss
durpath, generalized source spectrum
• Use figure from Erice paper showing reff
• TD vs RVT computations
93
94
95
96
97
Another Case
Study—PSA
vs R
98
The future (is here?):
Numerical simulations,
deterministic plus
stochastic
Continue to Rob Grave’s PPT slides
99